<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
Dismiss
Skip Navigation

Factorization using Difference of Squares

Factor binomials that are composed of subtracted perfect square terms

Atoms Practice
Estimated5 minsto complete
%
Progress
Practice Factorization using Difference of Squares
 
 
 
MEMORY METER
This indicates how strong in your memory this concept is
Practice
Progress
Estimated5 minsto complete
%
Practice Now
Turn In
Factorization using Difference of Squares

Factorization Using Difference of Squares 

When you learned how to multiply binomials we talked about two special products.

\begin{align*} \text{The sum and difference formula:} \quad (a + b)(a - b) & = a^2 - b^2\\ \text{The square of a binomial formulas:} \qquad \quad \ \ (a + b)^2 & = a^2 + 2ab + b^2\\ (a - b)^2 & = a^2 - 2ab + b^2\end{align*}

In this section we’ll learn how to recognize and factor these special products.

Factor the Difference of Two Squares

We use the sum and difference formula to factor a difference of two squares. A difference of two squares is any quadratic polynomial in the form \begin{align*}a^2 - b^2\end{align*}, where \begin{align*}a\end{align*} and \begin{align*}b\end{align*} can be variables, constants, or just about anything else. The factors of \begin{align*}a^2 - b^2\end{align*} are always \begin{align*}(a + b)(a - b)\end{align*}; the key is figuring out what the \begin{align*}a\end{align*} and \begin{align*}b\end{align*} terms are.

 

Factoring the Difference of Squares

1. Factor the difference of squares:

 

a) \begin{align*}x^2 - 9\end{align*}

Rewrite \begin{align*}x^2 - 9\end{align*} as \begin{align*}x^2 - 3^2\end{align*}. Now it is obvious that it is a difference of squares.

The difference of squares formula is:

\begin{align*} a^2 - b^2 = (a + b)(a - b)\end{align*}

Let’s see how our problem matches with the formula:

\begin{align*}x^2 - 3^2 = (x + 3)(x - 3)\end{align*}

The answer is:

\begin{align*}x^2 - 9 = (x + 3)(x - 3)\end{align*}

We can check to see if this is correct by multiplying \begin{align*}(x + 3)(x - 3)\end{align*}:

\begin{align*}& \quad \quad \ \ x + 3\\ & \underline{\;\;\;\;\;\;\;\;\;x - 3}\\ & \quad -3x - 9\\ & \underline{x^2 + 3x\;\;\;\;\;\;}\\ & x^2 + 0x - 9\end{align*}

The answer checks out.

Note: We could factor this polynomial without recognizing it as a difference of squares. With the methods we learned in the last section we know that a quadratic polynomial factors into the product of two binomials:

\begin{align*}(x\;\;\;\;)(x\;\;\;\;)\end{align*}

We need to find two numbers that multiply to -9 and add to 0 (since there is no \begin{align*}x-\end{align*}term, that’s the same as if the \begin{align*}x-\end{align*}term had a coefficient of 0). We can write -9 as the following products:

\begin{align*}& -9 = -1 \cdot 9 && \text{and} && -1 + 9 = 8\\ & -9 = 1 \cdot (-9) && \text{and} && 1 + (-9) = -8\\ & -9 = 3 \cdot (-3) && \text{and} && 3 + (-3) = 0 \qquad These \ are \ the \ correct \ numbers.\end{align*}

We can factor \begin{align*}x^2 - 9\end{align*} as \begin{align*}(x + 3)(x - 3)\end{align*}, which is the same answer as before. You can always factor using the methods you learned in the previous section, but recognizing special products helps you factor them faster.

b) \begin{align*}x^2 - 100\end{align*}

Rewrite \begin{align*}x^2 - 100\end{align*} as \begin{align*}x^2 - 10^2\end{align*}. This factors as \begin{align*}(x + 10)(x - 10)\end{align*}.

c) \begin{align*}x^2 - 1\end{align*}

Rewrite \begin{align*}x^2 - 1\end{align*} as \begin{align*}x^2 - 1^2\end{align*}. This factors as \begin{align*}(x + 1)(x - 1)\end{align*}.

2. Factor the difference of squares:

 

a) \begin{align*}16x^2 - 25\end{align*}

Rewrite \begin{align*}16x^2 - 25\end{align*} as \begin{align*}(4x)^2 - 5^2\end{align*}. This factors as \begin{align*}(4x + 5)(4x - 5)\end{align*}.

b) \begin{align*}4x^2 - 81\end{align*}

Rewrite \begin{align*}4x^2 - 81\end{align*} as \begin{align*}(2x)^2 - 9^2\end{align*}. This factors as \begin{align*}(2x + 9)(2x - 9)\end{align*}.

c) \begin{align*}49x^2 - 64\end{align*}

Rewrite \begin{align*}49x^2 - 64\end{align*} as \begin{align*}(7x)^2 - 8^2\end{align*}. This factors as \begin{align*}(7x + 8)(7x - 8)\end{align*}.

3. Factor the difference of squares:

 

a) \begin{align*}x^2 - y^2\end{align*}

\begin{align*}x^2 - y^2\end{align*} factors as \begin{align*}(x + y)(x - y)\end{align*}.

b) \begin{align*}9x^2 - 4y^2\end{align*}

Rewrite \begin{align*}9x^2 - 4y^2\end{align*} as \begin{align*}(3x)^2 - (2y)^2\end{align*}. This factors as \begin{align*}(3x + 2y)(3x - 2y)\end{align*}.

c) \begin{align*} x^2 y^2 - 1\end{align*}

Rewrite \begin{align*} x^2 y^2 - 1\end{align*} as \begin{align*}(xy)^2 - 1^2\end{align*}. This factors as \begin{align*}(xy + 1)(xy - 1)\end{align*}.

 

 

Examples

Factor the difference of squares:

 

Example 1

\begin{align*}x^4 - 25\end{align*}

Rewrite \begin{align*}x^4 - 25\end{align*} as \begin{align*}(x^2)^2 - 5^2\end{align*}. This factors as \begin{align*}(x^2 + 5)(x^2 - 5)\end{align*}.

Example 2

\begin{align*}16x^4 - y^2\end{align*}

Rewrite \begin{align*}16x^4 - y^2\end{align*} as \begin{align*}(4x^2)^2 - y^2\end{align*}. This factors as \begin{align*}(4x^2 + y)(4x^2 - y)\end{align*}.

Example 3

\begin{align*}x^2 y^8 - 64z^2\end{align*}

Rewrite \begin{align*}x^2 y^4 - 64z^2\end{align*} as \begin{align*}(xy^2)^2 - (8z)^2\end{align*}. This factors as \begin{align*}(xy^2 + 8z)(xy^2 - 8z)\end{align*}.

Review

Factor the following differences of squares.

  1. \begin{align*}x^2 - 4\end{align*}
  2. \begin{align*}x^2 - 36\end{align*}
  3. \begin{align*}-x^2 + 100\end{align*}
  4. \begin{align*}x^2 -400\end{align*}
  5. \begin{align*}9x^2 - 4\end{align*}
  6. \begin{align*}25x^2 - 49\end{align*}
  7. \begin{align*}9a^2 - 25b^2\end{align*}
  8. \begin{align*}-36x^2 + 25\end{align*}
  9. \begin{align*}4x^2 - y^2\end{align*}
  10. \begin{align*}16x^2 - 81y^2\end{align*}

Review (Answers)

To view the Review answers, open this PDF file and look for section 9.10. 

Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes
Please to create your own Highlights / Notes
Show More

Image Attributions

Explore More

Sign in to explore more, including practice questions and solutions for Factorization using Difference of Squares.
Please wait...
Please wait...